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This problem involves bacterial growth through binary fission, with periodic radiation that reduces the population. Here's a breakdown of the problem:

- Initial population: \( N_0 = 5 \times 10^3 \) bacteria.
- Generation time \( \tau = 30 \) seconds, meaning the population doubles every 30 seconds.
- At time \( t_r \), 90% of the population is destroyed (leaving 10% of the population).
- At time \( 3t_r \), 80% of the remaining population is destroyed (leaving 20% of the population).
- After the second irradiation, the remaining bacteria are five times the initial number.

We are tasked with finding \( t_r \).

### Step-by-step approach:
1. **Population Growth Formula**: Bacteria grow exponentially:
   \[
   N(t) = N_0 \times 2^{t/\tau}
   \]
   Where \( t \) is the time and \( \tau \) is the generation time.

2. **Population at \( t = t_r \)**: At time \( t_r \), the population is:
   \[
   N(t_r) = N_0 \times 2^{t_r/\tau}
   \]
   After irradiation, 90% of the bacteria are destroyed, leaving 10%:
   \[
   N(t_r \text{ after irradiation}) = 0.1 \times N(t_r)
   \]

3. **Population at \( t = 3t_r \)**: After more growth until \( 3t_r \):
   \[
   N(3t_r) = N_0 \times 2^{3t_r/\tau}
   \]
   After the second irradiation, 80% is destroyed, leaving 20%:
   \[
   N(3t_r \text{ after irradiation}) = 0.2 \times N(3t_r)
   \]
   The population after the second irradiation is given to be five times the initial number:
   \[
   N(3t_r \text{ after irradiation}) = 5 \times N_0
   \]

Now we can solve this system to find \( t_r \).

I'll calculate the necessary values for you.The value of \( t_r \) is approximately \( 46.44 \) seconds.

This means the first irradiation occurs at \( t_r = 46.44 \) seconds after the initial time, and the second irradiation occurs at \( 3t_r = 139.32 \) seconds.

Do you want more details on how these steps were computed, or any clarification?

Here are five related questions to deepen understanding:
1. How does binary fission affect bacterial population growth?
2. What would happen if a different percentage of the population were destroyed after each irradiation?
3. How does the generation time \( \tau \) influence the growth rate of the population?
4. Can this model apply to other exponential growth processes, like radioactive decay?
5. What mathematical tools can be used to analyze similar growth models?

**Tip**: Exponential growth and decay processes often follow similar mathematical structures, differing only in whether populations increase or decrease over time.

Considere un tipo de bacterias que se reproduce por bipartición, con un tiempo de generación τ = 30 [s]. Inicialmente (t = 0), se tienen 5 x 10^3 bacterias. En cierto instante tr se irradian las bacterias, destruyéndose el 90% de la población existente. Posteriormente, en el instante 3tr, se irradian nuevamente con lo que se destruye el 80% de la población. El número de bacterias que quedan justo después de la segunda irradiación es cinco veces el número inicial. Determine tr.

Learn how to calculate the time tr for a bacterial population undergoing binary fission, with irradiation destroying 90% and 80% of the population at different intervals. This problem explores exponential growth, population decay, and binary fission, suitable for undergraduate students in biology or applied mathematics.

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